Transcript Linear Functions

Functions & Relations
Vincent J Motto
University of Hartford
How would you use your calculator to
solve 52?
Input
Press:
5
Output
x2
25
• The number you entered is the input
number (or x-value on a graph). The input
values are called the domain.
• The result is the output number (or y-value
on a graph). Output values are the range.
• The x2 key illustrates the idea of a function.
A function is a relation that gives a single
output number for every valid input number.
A relation is a rule that produces one or more
output numbers for every valid input number.
There are many ways to represent relations:
•
•
•
•
•
Graph
Equation
Table of values
A set of ordered pairs
Mapping
These are all ways of
showing a relationship
between two variables.
A function is a rule that gives a single output number
for every valid input number.
To help remember & understand the definition:
Think of your input number, usually your
x-coordinate, as a letter.
Think of your output number, usually your
y-coordinate, as a mailbox.
A function is a rule that gives a single output number
for every valid input number.
Input number
Output number
Can you have one letter going to two different mail boxes?
Not a FUNCTION
A function is a rule that gives a single output number
for every valid input number.
Input number
Output number
Can you have two different letters going to one mail box?
Are these relations or functions?
x
1
2
3
4
y
5
6
7
Function
&
Relation
x
1
2
3
4
y
5
6
7
6
Are these relations or functions?
x
1
y
5
6
2
7
Not a Function but a
Relation
x
1
2
1
1
y
5
6
7
6
Are these relations or functions?
x
y
1
5
2
6
3
8
11
Not a function
But a relation
x
1
2
2
3
y
5
6
11
8
In words:
Double the number and add 3
As an equation:
y = 2x + 3
As a table of values:
x y
-2 -1
-1 1
0 3
1 5
These all
represent the
SAME function!
As a set of ordered pairs:
(-2, -1) (-1,1) (0,3) (1, 5) (2, 7) (3, 9)
Functional Notation
Functional Notation
• An equation that is a function may be
expressed using functional notation.
• The notation f(x) (read “f of (x)”)
represents the variable y.
Functional Notation Cont’d
Example:
y = 2x + 6 can be written as f(x) = 2x + 6.
Given the equation y = 2x + 6, evaluate when x = 3.
y = 2(3) + 6
y = 12
Functional Notation Con’t
For the function f(x) = 2x + 6, the notation f(3) means
that the variable x is replaced with the value of 3.
f(x) = 2x + 6
f(3) = 2(3) + 6
f(3) = 12
Evaluating Functions
Given f(x) = 4x + 8, find each:
1. f(2) = 4(2) + 8
= 16
2. f(a +1) = 4(a + 1) + 8
= 4a + 4 + 8
= 4a + 12
3. f(4a) = 4(-4a) + 8
= -16a+ 8
Evaluating More Functions
If f(x) = 3x  1, and g(x) = 5x + 3, find each:
1.
f(2) + g(3) = [3(2) -1] + [5(3) + 3]
= 6 - 1 + 15 + 3
= 23
2.
f(4) - g(-2) = [3(4) - 1] - [5(-2) + 3]
= 11 - (-7)
= 18
3.
3f(1) + 2g(2) = 3[3(1) - 1] + 2[5(2) + 3]
= 6 + 26
= 32
Ways of representing functions
In words:
Double the number and add 3
As an equation:
y = 2x + 3
As a table of values:
x y
-2 -1
-1 1
0 3
1 5
These are all ways of
showing a function
relationship between
two variables.
As a set of ordered pairs:
(-2, -1) (-1,1) (0,3) (1, 5) (2, 7) (3, 9)